Physic Labs

Problem 1

A particle of mass mm slides without friction from rest on a wedge of mass MM and incline angle α\alpha. The wedge rests on a smooth horizontal floor. Find the wedge's acceleration and the particle's acceleration relative to the wedge; gravity is gg downward. (a) State the assumptions and establish the governing equation for the requested quantity. (b) Solve it in terms of the given symbols, identifying the direction, sign, or physical nature of the result. (c) Check a simple limiting case and state when the model remains valid. Use SI units consistently, retain the symbols in the setup, and enforce all geometric constraints and initial conditions.
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
s¨=(M+m)gsin⁡αM+msin⁡2α,X¨=mgsin⁡αcos⁡αM+msin⁡2α\ddot s=\frac{(M+m)g\sin\alpha}{M+m\sin^2\alpha},\quad \ddot X=\frac{mg\sin\alpha\cos\alpha}{M+m\sin^2\alpha}
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Solving the two equations gives the result. The wedge accelerates toward the low end of the incline, while the particle accelerates down the slope relative to it.

problems.proof.pitfall. Do not discard an initial condition or stated constraint; an algebraic root outside the physical domain is not an answer.